Fractional rings of tangent spheres

Soddy’s hexlet consists of a ring of six spheres, tangent to each other consecutively around the ring, and another ring of three consecutively-tangent spheres, so that all the spheres in the first ring are tangent to all the spheres in the second ring. If you keep one ring fixed, you can rotate the other ring continuously, possibly changing the sizes of some of the spheres as they rotate but keeping the pattern of tangencies unchanged. Here’s a nice animation I found on Wikipedia, where the ring of six spheres rotates continuously while the other ring of three spheres (the central blue one and the two green planes, considered as degenerate spheres tangent at infinity) stays fixed. The larger red sphere is not part of this configuration and I don’t know why the author of this animation included it. We can describe the graph of tangencies of these nine spheres by using the join operation on graphs, which combines two graphs by adding edges from all vertices of one graph to all vertices…